Most mushroom compost is made from composted horse manure/straw bedding, along with a few other things. It ends up very dense humus, not something I would use alone and expect to work. It can also be laced with broadleaf herbicide residues and can vary from bag to bag/ batch to batch/ supplier to supplier/ region to region. The mushroom capitol of the world is a stone's throw away and the smell can be very potent, I am used to it but it is something you need to get used to...
Shiitakes, oysters and a few others are grown on artificial logs, made form sawdust, birdseed, and bran. That stuff seems like it would make better compost but must take at least a year or more to finish so I doubt it is used often, I know a hay farmer who spreads it on his fields after a short aging, they give it to him for free as long as he hauls it away.